Ideas from 'Nominalism and Substitutional Quantifiers' by Ruth Barcan Marcus [1978], by Theme Structure

[found in 'Philosophy of Logic: an anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21868-8]].

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
The nominalist is tied by standard semantics to first-order, denying higher-order abstracta
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
Anything which refers tends to be called a 'name', even if it isn't a noun
Nominalists see proper names as a main vehicle of reference
5. Theory of Logic / G. Quantification / 1. Quantification
Nominalists should quantify existentially at first-order, and substitutionally when higher
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Quantifiers are needed to refer to infinitely many objects
Substitutional semantics has no domain of objects, but place-markers for substitutions
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Maybe a substitutional semantics for quantification lends itself to nominalism
Substitutional language has no ontology, and is just a way of speaking
A true universal sentence might be substitutionally refuted, by an unnamed denumerable object
7. Existence / A. Nature of Existence / 3. Being / i. Deflating being
Is being just referent of the verb 'to be'?
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
Nominalists say predication is relations between individuals, or deny that it refers
9. Objects / A. Existence of Objects / 3. Objects in Thought
If objects are thoughts, aren't we back to psychologism?
9. Objects / F. Identity among Objects / 2. Defining Identity
Substitutivity won't fix identity, because expressions may be substitutable, but not refer at all